Dubai's royal family says Princess Latifa is "being cared for at home" after the BBC obtained videos in which she accused them of holding her hostage since she tried to flee in 2018.
"She continues to improve and we are hopeful she will return to public life at the appropriate time," said a statement from the royal family.
In the secretly recorded videos, the princess said she feared for her life.
The footage prompted global calls for a UN investigation.
The UN human rights agency has asked the United Arab Emirates (UAE) to provide proof that the princess is still alive.
But there was no video or photo released with the family's statement which could serve as proof of life, nor did it give any details about her condition.
US Secretary of State Antony Blinken told the BBC he would look into the case, adding: "We take human rights seriously, we will closely monitor the situation. Across the board, with adversaries, competitors, with partners and allies alike... we take human rights seriously and the president has put it at the heart of our foreign policy and countries can expect us to follow through on that."
The head of Human Rights Watch, Kenneth Roth, described the family's statement as a "cover story" and said no-one would believe Dubai's royals until Latifa had "the freedom to speak for herself".
The FreeLatifa campaign said similar statements from the royal family had been given in the past and could not be trusted.
In a statement, campaigners said they feared the Dubai authorities may have subjected her to "torture and drugging".
"It is now critically urgent that an independent team from the United Nations travel to Dubai and be able to visit Latifa immediately, and that they insist that Latifa is brought to safety in a country of her choosing," the FreeLatifa campaign said.
What did the Dubai royal family say?
The royal family's response came via the UAE embassy in London.
"In response to media reports regarding Sheikha Latifa, we want to thank those who have expressed concern for her wellbeing, despite the coverage which certainly is not reflective of the actual position," it said in a statement.
"Her family has confirmed that Her Highness is being cared for at home, supported by her family and medical professionals."
Princess Latifa's father, Sheikh Mohammed bin Rashid Al Maktoum, is one of the richest heads of state in the world, the ruler of Dubai and vice-president of the UAE.
How was Princess Latifa captured?
With the help of friends, Princess Latifa tried to flee Dubai to start a new life in February 2018.
"I'm not allowed to drive, I'm not allowed to travel or leave Dubai at all," she said in a video recorded just before her escape.
But days later, the princess was captured by commandos on a boat in the Indian Ocean. She was flown back to Dubai, where she has remained ever since.
Her father said he was acting in her best interests. Dubai and the UAE have previously said Princess Latifa was safe in the care of family.
But using a phone she said she was secretly given about a year after her capture, Princess Latifa recorded a number of videos over several months.
She recorded them in a bathroom as it had the only door she could lock. In the messages, she detailed how:
- she fought back against the soldiers taking her off the boat, "kicking and fighting" and biting one Emirati commando's arm until he screamed
- after being tranquillised, she lost consciousness as she was being carried on to a private jet, and didn't wake up until it landed in Dubai
- she was being held alone without access to medical or legal help in a villa with windows and doors barred shut, and guarded by police
On Friday the UN human rights office said it had approached the UAE's permanent mission in Geneva.
"We raised our concerns about the situation in light of the disturbing video evidence that emerged this week," Liz Throssell, spokeswoman for the UN High Commissioner for Human Rights, told a news briefing.
"We have asked for a proof of life - we have asked for further information."
Responding to the Dubai royal family's response, Human Rights Watch director Kenneth Roth said in a statement that Sheikh Mohammed "must have an odd definition of 'care'".
"When Latifa was able to communicate, we know he kept her locked up in prolonged solitary confinement - a form of torture," Roth said. "We can only imagine how that 'care' has degenerated now that his cruel conduct has been exposed."
He added: "No one will believe Sheikh Mohammed al-Maktoum about his feigned 'care' until he frees Latifa. Only when she is out from under his thumb will she have the freedom to speak for herself."